On the Regular inscribed Polygon of Twenty Sides. 267 



see that the last-mentioned function, viz. ( VO 2 -f 1 + 6) m , is an 

 integral of 



whence the transformed equation in 6 must be this very equa- 

 tion, that is, it must be the first equation. I have for shortness 

 used the particular integral ( \/ 1 -f x 2 + xf m ; but the reasoning 

 should have been applied, and it is in fact applicable, without 

 alteration, to the general integral 



C( •I+?+*r.+ C( VT+^ 2 -x) m . 

 There is of course no difficulty in a direct verification. Thus, 

 starting from the first equation, or equation in 0, the relation 

 iO = 2x 2 -\-l gives 



dy_ __ i_ dy d 2 y _ i d f i dy\ _ 1 /d 2 y 1 dy\ 



dd~4x'~dx' aW~'ix IxK^c doc) ~~ 16^ W 2 ~ x dxj' 

 l + <9 2 =-4^ 2 (l+# 2 ); 

 so that the equation becomes 



I (i .** ( *i __ I d J\ + l±M d JL _ m2?/ _ 



4 ll+f, .W #<fc/ + 4a? ^ my_a 

 Or multiplying by 4, 



,, <Py / 1-f-a? 2 l+2* 2 Wz/ . 2 . 



that 



t»+o3+'J-*-v-a 



the second equation. But the first method shows the reason 

 why the two forms are thus connected together. 



2 Stone Buildings, W.C., 

 February 19, 1862. 



XXXVII. Elementary Proof, that Eight Perimeters, of the Re- 

 gular inscribed Polygon of Twenty Sides, exceed Twenty-five 

 Diameters of the Circle. By Professor Sir William Rowan 

 Hamilton, LL.D., tyc* 



IT was proved by Archimedes that 71 perimeters, of a regular 

 polygon of 96 sides inscribed in a circle, exceed 223 dia- 

 meters; whence follows easily the well-known theorem, that 

 eight circumferences of a circle exceed twenty-five diameters, 

 or that 8 ir > 25. Yet the following elementary proof, that eight 

 perimeters of the regular inscribed polygon of twenty sides are 



* Communicated by the Author. 

 T 2 



